A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$, $K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On the line $OB$ a point $L$ is taken such that the lines $OA$ and $KL$ are parallel. Let $M$ be the intersection point of the circle $\omega$ circumscribed around triangle $KLB$, with line $AK$, with $M$ different from $K$. Prove that line $OM$ touches circle $\omega$.